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Metamath Proof Explorer

Definition df-en

Description: Define the equinumerosity relation. Definition of Enderton p. 129. We define ~ to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren . (Contributed by NM, 28-Mar-1998)

		Ref	Expression
	Assertion	df-en	$⊢ \approx = \{⟨x, y⟩ \| \exists f f : x \underset{1-1 onto}{⟶} y\}$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cen	$class \approx$
1		vx	$setvar x$
2		vy	$setvar y$
3		vf	$setvar f$
4	3	cv	$setvar f$
5	1	cv	$setvar x$
6	2	cv	$setvar y$
7	5 6 4	wf1o	$wff f : x \underset{1-1 onto}{⟶} y$
8	7 3	wex	$wff \exists f f : x \underset{1-1 onto}{⟶} y$
9	8 1 2	copab	$class \{⟨x, y⟩ \| \exists f f : x \underset{1-1 onto}{⟶} y\}$
10	0 9	wceq	$wff \approx = \{⟨x, y⟩ \| \exists f f : x \underset{1-1 onto}{⟶} y\}$